The false vacua of some potentials do not decay via Euclidean bounces. This typically happens for tunneling actions with a flat direction (in field configuration space) that is lifted by a perturbation into a sloping valley, pushing the bounce off to infinity. Using three different approaches we find a consistent picture for such decays. In the Euclidean approach the bottom of the action valley consists of a family of pseudo-bounces (field configurations with some key good properties of bounces except extremizing the action). The pseudo-bounce result is validated by minimizing a WKB action in Minkowski space along appropriate paths in configuration space. Finally, the simplest approach uses the tunneling action method proposed recently with a simple modification of boundary conditions.
We study the possibility of using machine learning for the calculation of the bounce action in quantum tunneling. Adopting supervised learning, we train neural network to give the bounce action from a given potential. It is found that, for one-dimensional tunneling, even a simple neural network performs at a percent level.
Quasiclassical methods are used to define dynamical tunneling times in models of quantum cosmological bounces. These methods provide relevant new information compared with the traditional treatment of quantum tunneling by means of tunneling probabilities. As shown here, the quantum dynamics in bounce models is not secure from reaching zero scale factor, re-opening the question of how the classical singularity may be avoided. Moreover, in the examples studied here, tunneling times remain small even for large barriers, highlighting the quantum instability of underlying bounce models.
We show that in Euclidean field theories that have bounce solutions, the bounce with the least action is the global minimum of the action in an open space of field configurations. A rigorous upper bound on the minimal bounce action can therefore be obtained by finite numerical methods. This sets a lower bound on the tunneling rate which, fortunately, is often the more interesting and useful bound. We introduce a notion of reduction which allows this bound to be computed with less effort by reducing complicated field theories to simpler ones.
I describe a class of oscillating bounce solutions to the Euclidean field equations for gravity coupled to a scalar field theory with multiple vacua. I discuss their implications for vacuum tunneling transitions and for elucidating the thermal nature of de Sitter spacetime.
The quantum decay of a metastable vacuum is exponentially suppressed by a tunneling action that can be calculated in the semi-classical approximation as the Euclidean action of a bounce that interpolates between the false and true phases. For multi-field potentials, finding the bounce is non-trivial due to its peculiar boundary conditions and the fact that the action at the bounce is not a minimum but merely a saddle point. Recently, an alternative tunneling action has been proposed that does not rely on Euclidean bounces and reproduces the standard result at its minimum. Here we generalize th…
We consider vacuum tunneling of a new kind where the false vacua are not translationally invariant, but have topological defects that break some of their translational symmetries. In the particular case where the topological defects are cosmic strings, we show the existence of an O(2) \times O(2) symmetric bounce configuration that allows a semi-classical estimate of the rate of cosmic string induced tunneling. A method of reduction is then suggested for simplifying the computation of the bounce action. Some phenomenological applications are described.
We study multi-field tunneling using exact solutions for additive potentials. We introduce a binomial potential with non-integer powers that could be considered a generalization of the $4D$ Fubini instanton potential. Using scaling arguments, we show that for multi-field potentials taller and wider barriers may still lead to a smaller bounce action.
The analytical bounce solution is derived in terms of the polygamma function in the Caldeira-Leggett's dissipative quantum tunneling model. The classical action for the bounce solution lies between the upper and lower bounds in the full range of $α$, where $α$ is a dissipation coefficient. The bounce peak point increases from 1 to 4/3 with increase of $α$. In spite of various nice features we have shown that the solution we have derived is not exact one by making use of the zero mode argument in the linearized fluctuation equation. However, our solution can be a starting point for approximate …
The periodic bounce configurations responsible for quantum tunneling are obtained explicitly and are extended to the finite energy case for minisuperspace models of the Universe. As a common feature of the tunneling models at finite energy considered here we observe that the period of the bounce increases with energy monotonically. The periodic bounces do not have bifurcations and make no contribution to the nucleation rate except the one with zero energy. The sharp first order phase transition from quantum tunneling to thermal activation is verified with the general criterions.
We study a class of oscillating bounce solutions to the Euclidean field equations for gravity coupled to a scalar field theory with two, possibly degenerate, vacua. In these solutions the scalar field crosses the top of the potential barrier $k>1$ times. Using analytic and numerical methods, we examine how the maximum allowed value of $k$ depends on the parameters of the theory. For a wide class of potentials $k_{\rm max}$ is determined by the value of the second derivative of the scalar field potential at the top of the barrier. However, in other cases, such as potentials with relatively flat…
Some false vacua do not decay via bounces. This usually happens when a flat direction of the tunneling action due to scale invariance is lifted to a sloping valley by a scale breaking perturbation, pushing the bounce off to infinity. We compare two types of alternative decay configurations that have been proposed recently to describe decay in such cases: pseudo-bounces and new instantons. Although both field configurations are quite similar, we find that the pseudo-bounce action is lower than the new instanton one and describes more faithfully the bottom of the action valley. In addition, pseu…
We evaluate both the vacuum decay rate at zero temperature and the finite temperature nucleation rate for the $(λϕ^4/4! + σϕ^6/6!)_{3D}$ model. Using the thin-wall approximation, we obtain the bounce solution for the model and we were also able to give the approximate eigenvalue equations for the bounce.
We derive the rate for transitions between de Sitter vacua by treating the field theory on the static patch as a thermal system. This reproduces the Coleman-De Luccia formalism for calculating the rate, but leads to a modified interpretation of the bounce solution and a different prediction for the evolution of the system after tunneling. The bounce is seen to correspond to a sequence of configurations interpolating between initial and final configurations on either side of the tunneling barrier, all of which are restricted to the static patch. The final configuration, which gives the initial …
Coleman tunneling in a general scalar potential with two non-degenerate minima is known to have an approximation in terms of a piecewise linear triangular-shaped potential with sharp 'kinks' at the place of the local minima. This approximate potential has a regime where the existence of the bounce solution needs the scalar field to 'wait' for some amount of Euclidean time at one of the 'kinks'. We discuss under which conditions a kink approximation of locally smooth 'cap' regions provides a good estimate for the bounce action.
A new formulation for obtaining the tunneling action for vacuum decay based on the so-called tunneling potential was developed recently. In the original derivation, the new action was obtained by requiring that its variation led to the correct equations of motion and the same value for the action as evaluated on the Euclidean bounce. We present an alternative derivation of the tunneling potential action via canonical transformations. We discuss both single-field and multi-field cases as well as the case with gravity. We also comment on the possible application of the new approach to the calcul…
It is shown that tunneling rates can be defined in terms of a false-vacuum effective action whose reality and convexity properties differ from those of the corresponding groundstate functional. The tunneling rate is directly related to the false-vacuum effective action evaluated at an extremal "quantum bounce". The Nielsen identities of the false-vacuum functional ensure that the rate remains independent of the choice of gauge-fixing. Our results are nonperturbative and clarify issues related with convexity and radiative corrections.
The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of $N$ scalars with a random potential of fourth order. Here we show that the scaling of the typical minimal bounce action with $N$ is readily understood, and differs from statements in the literature. We discuss the extension to more realistic landscape models.
An alternative approach to the calculation of tunneling actions, that control the exponential suppression of the decay of metastable phases, is presented. The new method circumvents the use of bounces in Euclidean space by introducing an auxiliary function, a tunneling potential $V_t$ that connects smoothly the metastable and stable phases of the field potential $V$. The tunneling action is obtained as the integral in field space of an action density that is a simple function of $V_t$ and $V$. This compact expression can be considered as a generalization of the thin-wall action to arbitrary po…
We present exact bounce solutions and amplitudes for tunneling in i) a piecewise linear-quartic potential and ii) a piecewise quartic-quartic potential. We cross check their correctness by comparing with results obtained through the thin-wall approximation and with a piecewise linear-linear potential. We briefly comment on applications in cosmology.
We clarify issues of convexity, gauge-dependence and radiative corrections in relation to tunneling rates. Despite the gauge dependence of the effective action at zero and finite temperature, it is shown that tunneling and nucleation rates remain independent of the choice of gauge-fixing. Taking as a starting point the functional that defines the transition amplitude from a false vacuum onto itself, it is shown that decay rates are exactly determined by a non-convex, false vacuum effective action evaluated at an extremum. The latter can be viewed as a generalized bounce configuration, and gaug…
Using numerical and analytic methods, we study quantum tunneling from a Minkowski false vacuum to an anti-de Sitter true vacuum. Scanning the parameter space of theories with quartic and non-polynomial potentials, we find that for any given potential tunneling is completely quenched if gravitational effects are made sufficiently strong. For potentials where $ε$, the energy density difference between the vacua, is small compared to the barrier height, this occurs in the thin-wall regime studied by Coleman and De Luccia. However, we find that other potentials, possibly with $ε$ much greater than…
We present a simple and intuitive description of both, the Schwinger effect and false vacuum decay through bubble nucleation, as tunneling problems in one-dimensional relativistic quantum mechanics. Both problems can be described by an effective potential that depends on a single variable of dimension length, which measures the separation of the particles in the Schwinger pair, or the radius of a bubble for the vacuum decay. We show that both problems can be described as tunneling in one-dimensional quantum mechanics if one interprets this variable as the position of a relativistic particle wi…
A positive definite formulation of vacuum decay with reduced symmetry
Instituto de Física Teórica IFT-UAM/CSIC（理论物理研究所 IFT-UAM/CSIC）；Department of Physics, Graduate School of Science, Kobe University（神户大学理学研究科物理学系）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器 DESY）；Department of Physics, Osaka Metropolitan University（大阪公立大学物理学系）；Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan University（大阪公立大学大阪中央先进数学研究所 (OCAMI)）
We discuss new possible tunneling processes in the presence of gravity. We formulate quantum tunneling using the Wheeler-deWitt canonical quantization and the WKB approximation. The distinctive feature of our formulation is that it accommodates the coexistence of Euclidean and Lorentzian evolution. It opens up a new possibility of quantum tunneling; e.g. a bubble wall itself tunnels the potential barrier pulling the field nearby, where the wall region experiences the Euclidean evolution while the other regions experience the Lorentzian evolution simultaneously. We execute numerical analysis an…
A Fubini instanton is a bounce solution which describes the decay of a vacuum state located at the top of the tachyonic potential {\it via} the tunneling without a barrier. We investigate various types of Fubini instantons of a self-gravitating scalar field under a tachyonic quartic potential. With gravity taken into account, we show there exist various types of unexpected solutions including oscillating bounce solutions. We present numerically oscillating Fubini bounce solutions in anti-de Sitter and de Sitter spaces. We construct the parametric phase diagrams of the solutions, which is the e…
We consider the vacuum decay of the flat Minkowski space to an anti-de Sitter space. We find a one-parameter family of potentials that allow exact, analytical instanton solutions describing tunneling without barriers in the presence of gravity. In the absence of gravity such instantons were found and discussed by Lee and Weinberg more than a quarter of a century ago. The bounce action is also analytically computed. We discuss possible implications of these new instantons to cosmology in the context of the string theory landscape.
The decay rate of a false vacuum is determined by the minimal action solution of the tunnelling field: bounce. In this Letter, we focus on models with scalar fields which have a canonical kinetic term in $N(>2)$ dimensional Euclidean space, and derive an absolute lower bound on the bounce action. In the case of four-dimensional space, we show the bounce action is generically larger than $24/λ_{\rm cr}$, where $λ_{\rm cr} \equiv {\rm max} [ -4V(ϕ) /|ϕ|^4] $ with the false vacuum being at $ϕ=0$ and $V(0)=0$. We derive this bound on the bounce action \textit{without solving the equation of motion…
We study the semiclassical fluctuation problem around bounce solutions for a self-interacting scalar field in curved space. As in flat space, the fluctuation problem separates into partial waves labeled by an integer l, and we determine the large l behavior of the fluctuation determinants, a quantity needed to define a finite fluctuation prefactor. We also show that while the Coleman-De Luccia bounce solution has a single negative mode in the l=0 sector, the oscillating bounce solutions also have negative modes in partial waves higher than the s-wave, further evidence that they are not directl…
Quantum cosmology uses a wave function to model the universe, but finding solutions for this poses a problem as it is difficult to define the boundary conditions or identify the correct path for a path integral. We begin the discussion by going over various proposals and look at how bubble universe nucleation can be used as an analogy for the tunneling wave function. We review how the Hartle-Hawking wave function and tunneling wave functions are equivalent. This leads into how the transition between the decelerated and accelerated expansion is formulated as a bounce in connection space. This i…
The tunneling potential method to calculate the action for vacuum decay is an alternative to the Euclidean bounce method that has a number of attractive features. In this paper we extend the formalism to general spacetime dimension $d>2$ and use it to give simple proofs of several results. For Minkowski or Anti de Sitter false vacua, we show that gravity or higher barriers increase vacuum lifetime and describe a very clean picture of gravitational quenching of vacuum decay. We also derive the thin-wall limit of the action, show how detailed balance for dS to dS transitions works in the new for…
We provide an $F(R)$ gravity description of a $Λ$CDM bouncing model, without the need for matter fluids or for cosmological constant. As we explicitly demonstrate, the two cosmological eras that constitute the $Λ$CDM bouncing model, can be generated by $F(R)$ gravity which can lead to accelerating cosmologies. The resulting F(R) gravity has Einstein frame inflationary properties that have concordance to the latest Planck observational data. Both the $F(R)$ gravity stability properties are thoroughly investigated and also, the gravitational particle production, a feature necessary for the viabi…
We consider the decay of "false kinks," that is, kinks formed in a scalar field theory with a pair of degenerate symmetry-breaking false vacua in 1+1 dimensions. The true vacuum is symmetric. A second scalar field and a peculiar potential are added in order for the kink to be classically stable. We find an expression for the decay rate of a false kink. As with any tunneling event, the rate is proportional to $\exp(-S_E)$ where $S_E$ is the Euclidean action of the bounce describing the tunneling event. This factor varies wildly depending on the parameters of the model. Of interest is the fact t…
Based on the gradient flow, we propose a new method to determine the bounce configuration for false vacuum decay. Our method is applicable to a large class of models with multiple fields. Since the bounce is a saddle point of an action, a naive gradient flow method which minimizes the action does not work. We point out that a simple modification of the flow equation can make the bounce its stable fixed point while the false vacuum configuration an unstable one. Consequently, the bounce configuration can be obtained simply by following the flow without a careful choice of an initial configurati…
False vacuum states are metastable in quantum field theories, and true vacuum bubbles can be nucleated due to the quantum tunneling effect. It was recently suggested that an evaporating black hole (BH) can be a catalyst of bubble nucleations and dramatically shortens the lifetime of the false vacuum. In particular, in the context of the Standard Model valid up to a certain energy scale, even a single evaporating BH may spoil the successful cosmology by inducing the decay of our electroweak vacuum. In this paper, we reinterpret catalyzed vacuum decay by BHs, using an effective action for a thin…
Metastable states decay at zero temperature through quantum tunneling at an exponentially small rate, which depends on the Coleman-de Luccia instanton, also known as bounce. In some theories, the bounce may not exist or its on-shell action may be ill-defined or infinite, thus hindering the vacuum decay process. In this paper, we test this possibility in modified theories of gravity interacting with a real scalar field. We consider an Einstein-Hilbert term with a non-minimally coupled scalar field and a quadratic Ricci scalar contribution. To tackle the problem we use a new analytic method, wit…
We study Coleman-de Luccia tunneling in some detail. We show that, for a single scalar field potential with a true and a false vacuum, there are four types of tunneling, depending on the properties of the potential. A general tunneling process involves a combination of thermal (Gibbons-Hawking temperature) fluctuation part way up the barrier followed by quantum tunneling. The thin-wall approximation is a special limit of the case (of only quantum tunneling) where inside the nucleation bubble is the true vacuum while the outside reaches the false vacuum. Hawking-Moss tunneling is the (only ther…
The bouncing evolution of an universe in Loop Quantum Cosmolgy can be described very well by a set of effective equations, involving a function $sin \; x$. Recently, we have generalised these effective equations to $(d + 1)$ dimensions and to any function $f(x) \;$. Depending on $f(x) \;$ in these models inspired by Loop Quantum Cosmolgy, a variety of cosmological evolutions are possible, singular as well as non singular. In this paper, we study them in detail. Among other things, we find that the scale factor $a(t) \; \propto \; t^{ \frac {2 q} {(2 q - 1) \; (1 + w) d}} \;$ for $f(x) = x^q \;…
In this letter, we consider cosmological solutions of bimetric theory without assuming that only one metric is coupled to gravity. We conclude that any cosmology can be described by fixing the matter content of the space that we are not inhabiting. On the other hand, we show that some conclusions can still be extracted independently of the matter content filling both spaces. In particular, we can conclude the occurrence of some extremality events in one universe if we know that they take place in the other space.
GUP Effective Metric Without GUP: Implications for the Sign of GUP Parameter and Quantum Bounce
Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University（扬州大学物理科学与技术学院引力与宇宙学中心）；Shanghai Frontier Science Center for Gravitational Wave Detection, School of Aeronautics and Astronautics, Shanghai Jiao Tong University（上海交通大学航空航天学院上海引力波探测前沿科学中心）
In flat space and at finite temperature, there are two regimes of false vacuum decay in quantum field theory. At low temperature, the decay proceeds through thermally-assisted tunneling described by periodic Euclidean solutions -- bounces -- with non-trivial time dependence. On the other hand, at high temperature the bounces are time-independent and describe thermal jumps of the field over the potential barrier. We argue that only solutions of the second type are relevant for false vacuum decay catalyzed by a black hole in equilibrium with thermal bath. The argument applies to a wide class of …
We recently showed how it is possible to use a cubic Galileon action to construct classical cosmological solutions that enter a contracting null energy condition (NEC) violating phase, bounce at finite values of the scale factor and exit into an expanding NEC-satisfying phase without encountering any singularities or pathologies. A drawback of these examples is that singular behavior is encountered at some time either just before or just after the NEC-violating phase. In this Letter, we show that it is possible to circumvent this problem by extending our method to actions that include the next…
We investigate possible geometries allowing transitions from a black hole to a white hole spacetime, by placing a space-like thin shell between them. Such proposals have been advanced recently to account for singularity-resolution in black-hole spacetimes. This space-like shell can be extended to be outside the event horizon and, thereby, reproduce some of the features of these proposals. On the other hand, if the space-like shell is confined fully within the horizon, then it results in a bounce near a space-like singularity inside the black hole. For both cases, the null energy condition is n…
Hawking radiation is studied for arbitrary scalars, fermions, and spin-1 bosons, using a tunneling approach, to every order in $\hbar$ but ignoring back-reaction effects. It is shown that the additional quantum terms yield no new contribution to the Hawking temperature. Indeed, it is found that the limit of small $\hbar$ in the standard quantum WKB approximation is replaced by the near-horizon limit in the gravitational WKB approach.
Ekpyrotic bouncing cosmologies have been proposed as alternatives to inflation. In these scenarios, the universe is smoothed and flattened during a period of slow contraction preceding the bounce while quantum fluctuations generate nearly scale-invariant super-horizon perturbations that seed structure in the post-bounce universe. An analysis by Tolley and Wesley (2007) showed that, for a wide range of ekpyrotic models, generating a scale-invariant spectrum of adiabatic or entropic fluctuations is only possible if the cosmological background is unstable, in which case the scenario is highly sen…
